My biggest tip for anyone struggling: play the entire fight regardless of how early you take your first hit. This allows you to become familiar with the whole fight, not just the first rounds. It's hard to avoid chaos in the 4th and final round, so unless you have supernatural instincts or extreme 3D platforming chops, you're going to need a bit of practice to be able to parse and navigate the chaos. And, annoyingly, you won't get that practice if you start over every time you get dinged in Round 2, say.
ROUND 1
Before the chickens spawn, position yourself in the middle of the grid. The first chicken might be the most dicey one, since you need to avoid its plant spawns without then running into one of the other chickens by accident. Use your jumps of the heads of the chicken to preemptively get out of the way of the plant spawns.
I like to start with the top right chicken, which gives me a decent overview over the rest of the board when I'm positioning myself to avoid the plant-smacks. Try to only aggro only two plants at a time, that way it's easier to stay in control of the board. Like I said, I start with the top right chicken, then go top left, bottom right and finally bottom left.
You have all the time in the world with this first one, so it's a good place to practice if you're struggling to nail the positioning when you're jumping on enemies.
ROUND 2
Move to the middle of the grid before the first four spawns. Remaining airborne here after jumping on the first enemy is the safest way of staying control of the board, so use your double jumps to increase your airtime while you're changing directions and jumping from enemy to enemy. The posititions of the second and third round of spawns are slightly random, and it can be a bit difficult to track the exact spawn locations based on the pink spawn-cloud thingies. Do your best to stay out of the positions where it looks like the spawns are going to happen, and then start tail-spinning so that you'll knock out any enemy that spawns too close to you.
As for the flame cannons, I've found it's easier to track the position of the towers than it is to keep track of all the individual fireballs. Ultimately, staying airborne is the best way to avoid the fireballs, so practice your jumps and airmovements. Oh, and mind the little hop that the worms make right after they spawn, it can catch you out if you don't account for it.
ROUND 3
The trick here is to move decisively and quickly. Get the enemies off the board as soon as possible is imperative, so it doesn't become a mess of wasps, flame beams and projectiles. Getting quick and decisive enough might take practice, though (see the advice in the intro).
On the first spawn, move to the left out of the corner, and then jump on the bottom right wasp before it shoots. You can either then rush to jump on the bottom left enemy, or wait until it fires, and tailspin its projectile back. Move back to the bottom right corner, avoiding the flame beams, and then advance on the upper right enemy. If it fires on you before you reach it, tailspin its projectile. If not, jump on it. Same approach with the top left enemy.
Second batch of spawns, you should be able to finish off all enemies with returned projectiles, which means you shouldn't need to be jumping across any of the spikes. Just stay in the bottom half of the grid, be mindful of the flame beams, and you should be fine. Start with the bottom right enemy, then take care of the bottom left. Finally, move a little to the right, positioning yourself between the top two enemies, where you should be able to return both their projectiles from relative safety (remember the flame beams, though!).
Third batch, I recommend taking out all the enemies by jumping on them. The limited floor space and flame beams make it a bit dicey to stand around waiting to return projectiles, so jumping is the safest strategy. Start with the closest enemy (bottom right), jumping right off it to the top right enemy without touching the ground. You can keep jumping from enemy to enemy, but I prefer to take my time a little with the last two enemies. Take a moment to wait for the flame beams to move, and jump for the clear floor tile diagonally down from the top left enemy. Jump on it (or return its projectile if the timing is right), and beware that the last enemy doesn't get the drop on you with a projectile from below you, off-screen. Jump to the clear floor tile diagonally up to the left from the last enemy, and then jump kill or return its projectile.
ROUND 4
Like I said above, this is chaotic. My best tip is to spam the tailspin as much as you can while still maintaining control over your jumps; that way, if you misjudge a jump and sidle into an enemy, chances are you'll stun them instead of getting hit yourself. In my opinion, the biggest threat in this fight is the flame sprite that spawns after you kill the golems, and then homes in on you. This thing is aggravating to deal with if there's too much else happening on the board, so, in each batch, try to kill the wasp enemies before you take out the golem and release the sprite. Since the flame sprite spawns low to the ground, I recommend that you deliver the killing blow (the golems take three hits to kill) with a jump to the golem's head, making sure that you create as much distance off the flame sprite as you can off that final headbop.
The diciest section in this round are the in-between-phases, where you've killed all the enemies, you're avoiding the flame sprite, and then the next batch of enemies spawn in while that dang flame sprite is still on the board, and the flame beams are repositioning and revving up again. I'm not going to lie, it's an annoying, extra chaotic time. The placement of the rotating flame beams is randomised, but, thankfully the enemy spawns are fixed, so with a bit of practice, you'll know roughly which area to go to when avoiding the flame sprite, so the enemies don't spawn on top of you.
Good luck!